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For each integer N g> 1 it follows from (13.5.7) that

N                      N

\n=l    /        n=l


ow apply (12,5.7) to the increasing sequence of partial sums

n= 1

If (/n) is any sequence of mappings of X into R, the mapping
x h^lim inf /(*)      (resp. x h^lim sup/n(x))

n-*oo                                                      n-*oo

j defined for all x e X. It is denoted by lim inf/w (resp. lim sup/n).

H-+QO                             -*oo

13.5,9)       (Fatou's lemma)   If(fn) is any sequence of functions ^0, then
M*(lim inf/.) glim inf /im

\   i-oo        /           n-+oo

For each n ^ 1, let gn = inf (fn+p). Clearly we have n*(gn] ^ inf ^*(/n+p

1>0                                                                       p^O

nd ^n ^0; since the sequence (0rB) is increasing and liminfyj, = sup^nj it
ollows from (13.5.7) that

^*/lim inf/n\ = sup v*(gn} ^ sup (inf jU*(

\   n-oo         Jn                  n     \p^0

If/: X ~> K is any mapping, we define /**(/) =  /**(-/); this number is
ailed the /0vtw integral of /with respect to the measure /i. All the properties
.bove which were proved for the upper integral can be immediately translated
tito properties of the lower integral. In particular, if we put  J = ^ (or
^(X)), then y is the set of all upper semicontinuous functions on X which are
Bounded above by a function belonging to 5TR(X) (which implies that they do
lot take the value + oo). For all functions /e f, we have

ind for all/: X -* R we have

=    sup

[1 3.5.10)   Iff is any mapping ofX into I, then /**(/) <; /i*(/). otherwise there is